24
Mathematical Analysis
REGGIO DI CALABRIA
Overview
Date/time interval
Syllabus
Course Objectives
The course aims at providing a suitable knowledge of the principles and methods of the theory of functions of complex variables and of Z transform and a suitable knowledge of random systems of interest in computer sciences, electronics and communications systems.
Expected learning outcomes (according to Dublin Descriptors)
Knowledge and understanding: Upon passing the exam, the student will know the fundamental principles of the theory of functions of a complex variable, which provides a range of essential mathematical tools for solving engineering problems, as well as the Z-transform, used in numerical signal processing. They will also be familiar with the main queuing models.
Applying knowledge and understanding: Upon passing the exam, the student will be able to apply the theoretical knowledge acquired to problems related to the resolution of integrals, recurrence-defined sequences, and queuing theory.
Making judgements: : To pass the exam, the student must be able to recognize the situations and problems where the basic techniques of the theory of functions of a complex variable, the Z-transform, and birth-death processes can be applied.
Communication skills: Upon passing the exam, the student will be able to communicate the knowledge acquired using appropriate technical-scientific language to both specialist and non-specialist audiences.
Lifelong learning skills: Upon passing the exam, the student will be capable of independently deepening their acquired knowledge and applying it to understand new topics to be addressed in the continuation of their studies.
Course Prerequisites
knowledge of Calculus I, Calculus II and Probability Theory
Teaching Methods
Lectures and exercises
Assessment Methods
LM-27
The exam consists of two parts: a written test (two hours and thirty minutes) and an oral test. The written test aims to assess the student's ability to solve exercises on the applications of the Residue Theorem, the Z-Transform, and queuing models. Maximum score: 30/30. Passing the written test allows access to the oral test.
The oral test is designed to assess the level of knowledge and understanding of the course content, evaluate judgment autonomy, learning ability, and communication skills. The oral test consists of a discussion of the written test and theoretical questions about the course content. Maximum score: 30/30. The final grade for the exam is determined by considering both the written and oral tests.
Evaluation Criteria:
- 30 cum laude: complete, in-depth, and critical knowledge of the topics, excellent command of language, comprehensive and original interpretative skills, full ability to autonomously apply knowledge to solve proposed problems;
- 28 - 30: complete and in-depth knowledge of the topics, excellent command of language, effective interpretative skills, capable of autonomously applying knowledge to solve proposed problems;
- 24 - 27: good knowledge of the topics with a solid degree of mastery, good command of language, accurate and confident interpretative skills, good ability to correctly apply most knowledge to solve proposed problems;
- 20 - 23: adequate knowledge of the topics but limited mastery, satisfactory command of language, correct interpretative ability, more than sufficient ability to autonomously apply knowledge to solve proposed problems;
- 18 - 19: basic knowledge of the topics, basic knowledge of technical language, sufficient interpretative ability, sufficient ability to apply the basic knowledge acquired;
- <18 Unsatisfactory: the student does not possess an acceptable knowledge of the topics covered in the course.
LM-29
The exam consists of a written test (two hours) with exercises and theoretical questions. The test aims to assess the student's ability to solve exercises on the applications of the Residue Theorem and the level of knowledge and understanding of the course content, to evaluate judgment autonomy, learning ability, and communication skills.
Evaluation Criteria:
- 30 cum laude: complete, in-depth, and critical knowledge of the topics, excellent command of language, comprehensive and original interpretative skills, full ability to autonomously apply knowledge to solve proposed problems;
- 28 - 30: complete and in-depth knowledge of the topics, excellent command of language, effective interpretative skills, capable of autonomously applying knowledge to solve proposed problems;
- 24 - 27: good knowledge of the topics with a solid degree of mastery, good command of language, accurate and confident interpretative skills, good ability to correctly apply most knowledge to solve proposed problems;
- 20 - 23: adequate knowledge of the topics but limited mastery, satisfactory command of language, correct interpretative ability, more than sufficient ability to autonomously apply knowledge to solve proposed problems;
- 18 - 19: basic knowledge of the topics, basic knowledge of technical language, sufficient interpretative ability, sufficient ability to apply the basic knowledge acquired;
- <18 Unsatisfactory: the student does not possess an acceptable knowledge of the topics covered in the course.
Texts
G.Di Fazio, M.Frasca, Metodi Matematici per l'Ingegneria, Monduzzi Editore
G.Teppati, Esercitazioni di Analisi Matematica III, Progetto Leonardo.
F.S. Hillier and G.J.Lieberman, Introduzione alla Ricerca Operativa, Collana di Matematica e Statistica Franco Angeli.
Contents
Mathematical Methods for Engineering 6 CFU (LM-27)
Complex valued functions of complex variable. Complex valued functions. Elementary functions of complex variable. Cauchy-Riemann equations. Holomorphic functions. Reminds about planar curves and line integrals. Complex integration. Cauchy-Goursat Theorem. Cauchy integral formulas. Reminds about power series. Analytic functions. Taylor series. Taylor Theorem. Laurent series. Laurent expansions. Isolated singular points and classification. Residues. Calculus of residues. Singularity at infinity and classification. Residue Method for Partial Fractions Decomposition. Residue Theorem and Corollary. Applications of Residue Theorem. (3 CFU)
Z transform: definition, properties and examples. Applications of Z transform to recurrence sequences and difference equations.(1 CFU)
Fundamentals of Queue Theory. Introduction to Queue Theory: description of the Queue Problem. Characteristics of Queue Processes. Kendall notation. Measuring System Performance. Little’s Formula. Poisson Process and the Exponential Distribution. Markovian property of the exponential distribution. Stochastic processes and Markov chains. Chapman-Kolmogorov equations. Flow balance equations. Steady state probability distribution. Birth Death Processes. Single-Server Queues (M/M/1). Queues with Truncation M/M/1/k. M/M/1 State-Dependent Service. M/M/1/∞/R - Finite Source Queues. Queues M/M/s. Queues with Truncation M/M/s/k. Coda M/M/∞. Queues M/Ek/1, with Erlang Service. Queues M/G/1 General Service, Single Server. Priority Queues. (2 CFU)
Mathematical Methods for Engineering 3 CFU (LM-29)
Complex valued functions of complex variable. Complex valued functions. Elementary functions of complex variable. Cauchy-Riemann equations. Holomorphic functions. Reminds about planar curves and line integrals. Complex integration. Cauchy-Goursat Theorem. Cauchy integral formulas. Reminds about power series. Analytic functions. Taylor series. Taylor Theorem. Laurent series. Laurent expansions. Isolated singular points and classification. Residues. Calculus of residues. Singularity at infinity and classification. Residue Method for Partial Fractions Decomposition. Residue Theorem and Corollary. Applications of Residue Theorem. (3 CFU)
More information
Team Code: op2mgai